@Article{ATA-30-1,
author = {},
title = {Complex and $p$-Adic Meromorphic Functions $f'P'(f)$, $g'P'(g)$ Sharing a Small Function},
journal = {Analysis in Theory and Applications},
year = {2014},
volume = {30},
number = {1},
pages = {51--81},
abstract = {<p style="text-align: justify;">Let $\mathbb{K}$ be a complete algebraically closed $p$-adic field of characteristic zero. We apply results in algebraic geometry and a new Nevanlinna theorem for $p$-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on $\mathbb{K}$ and on $\mathbb{C}$. Let $P$ be a polynomial of uniqueness for meromorphic functions in $\mathbb{K}$ or $\mathbb{C}$ or in an open disk. Let $f$, $g$ be two transcendental meromorphic functions in the whole field $\mathbb{K}$ or in $\mathbb{C}$ or meromorphic functions in an open disk of $\mathbb{K}$ that are not quotients of bounded&nbsp; analytic functions. We show that if $f&#39;P&#39;(f)$ and $g&#39;P&#39;(g)$ share a small function $\alpha$ counting multiplicity, then $f=g$, provided that the multiplicity order of zeros of $P&#39;$ satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities $n\geq k+2$ or $n\geq k+3$ used in previous papers by Hypothesis (G). In the $p$-adic context, another consists of giving a lower bound for a sum of $q$ counting functions of zeros with $(q-1)$ times the characteristic function of the considered meromorphic function.</p>},
issn = {1573-8175},
doi = {https://doi.org/10.4208/ata.2014.v30.n1.4},
url = {https://global-sci.com/article/73976/complex-and-p-adic-meromorphic-functions-fpf-gpg-sharing-a-small-function}
}